Nationalrail.co.uk is probably your best source.
As an ex-Londoner, I remember it has only been ten years since the first rail connection between central London and it's largest airport, Heathrow. Due to Heathrow's geographical location, I doubt there are any direct connections. Even when connecting through London Paddington railway station, you will probably need to take a taxi or the 'tube' (subway network) to change to the railway station which services your eventual destination. If I remember correctly, Paddington station (where the Heathrow express arrives) only services the West of England. (I remember it being; Swindon, Bath, Bristol, etc).... For example, to get to Southhampton or Kent (south of London), I imagine you will need to go to Victoria or another station that services the South of England. To go to Edinburgh, I know you have to go all the way to King's Cross Railway station. Each rail station services a geographical area of the country, simmilar to Paris and the French rail network.
If I'm correctly informed, London's politicians are still arguing about having a railway line circumnavigating the capital city... 100+ years since they invented the thing!
Good luck! All I can say is that it will be a long trek! Maybe spend a night in London first! almost 8 years ago